Interval Computations Gauging Expert . . . Processing Expert . . . - PowerPoint PPT Presentation

Need for Data Processing Need for Expert Estimates Need to Take . . . Interval Computations Gauging Expert . . . Processing Expert . . . in Metrology Interval and Fuzzy . . . Case Study: Heat Meter Vladik Kreinovich 1 , Konstantin Semenov 2 , and Acknowledgments Gennady N. Solopchenko 2 Home Page Title Page 1 Department of Computer Science, University of Texas at El Paso El Paso, TX 79968, USA, vladik@utep.edu ◭◭ ◮◮ 2 Peter the Great St. Petersburg Polytechnic University ◭ ◮ 29 Polytechnicheskaya str., St. Petersburg, 195251, Russia Page 1 of 13 semenov.k.k@iit.icc.spbstu.ru, g.n.solopchenko@iit.icc.spbstu.ru Go Back Full Screen Close Quit

Need for Data Processing 1. Need for Data Processing Need for Expert Estimates Need to Take . . . • In many engineering situations, we need to make deci- Gauging Expert . . . sions. Processing Expert . . . • Some of these decisions are made by humans, some by Interval and Fuzzy . . . automatic control systems. Case Study: Heat Meter Acknowledgments • The decisions y are based on the valued of the relevant Home Page quantities x 1 , . . . , x n : y = f ( x 1 , . . . , x n ). Title Page • Ideally, the values x i should come from measurement. ◭◭ ◮◮ • However, in many cases, we also need to use expert ◭ ◮ estimates. Page 2 of 13 • This is typical, e.g., in inverse problems, which are, in general, ill-defined. Go Back Full Screen Close Quit

Need for Data Processing 2. Need for Expert Estimates Need for Expert Estimates Need to Take . . . • For example, we may be interested in the value x ( t ), Gauging Expert . . . but sensors only measure averages Processing Expert . . . � t + ε � Interval and Fuzzy . . . K ( t − t ′ ) · x ( t ′ ) dt and x av ( t ) = K ( τ ) dτ = 1 . Case Study: Heat Meter t − ε Acknowledgments • To make these problems well-defined, we need to add Home Page prior information – which comes from experts. Title Page • For example, in measuring x ( t ), the experts can give ◭◭ ◮◮ us the upper bound M on the rate of change | ˙ x ( t ) | . ◭ ◮ • In this case, | x ( t ) − x av ( t ) | ≤ M · ε . Page 3 of 13 • Both measurement results and expert estimates come Go Back with uncertainty. Full Screen Close Quit

Need for Data Processing 3. Need to Take Uncertainty into Account Need for Expert Estimates Need to Take . . . • Measurements are never absolutely accurate. Gauging Expert . . . • The measurement result � x is, in general, different from Processing Expert . . . the actual value x of the corresponding quantity. Interval and Fuzzy . . . Case Study: Heat Meter • Ideally, we should know the probability distribution for def Acknowledgments the measurement error ∆ x = � x − x . Home Page • However, in most practical cases, all we know is the Title Page upper bound ∆ on the measurement error: | ∆ x | ≤ ∆. ◭◭ ◮◮ • In this case, once we have a measurement result � x , all ◭ ◮ we know about the actual value x is that Page 4 of 13 x ∈ [ � x − ∆ , � x + ∆] . Go Back • Expert estimates are also imprecise. Full Screen Close Quit

Need for Data Processing 4. Gauging Expert Uncertainty Need for Expert Estimates Need to Take . . . • Ideally, we should view each expert as a measuring Gauging Expert . . . instrument: Processing Expert . . . – we compare expert estimates and measurement re- Interval and Fuzzy . . . sults, and Case Study: Heat Meter – we get a probability distribution for the estimation Acknowledgments error ∆ x = � x − x . Home Page Title Page • In practice, we rarely have enough samples to make statistically meaningful estimates. ◭◭ ◮◮ • A reasonable way to describe expert uncertainty is to ◭ ◮ ask the expert to estimate, Page 5 of 13 – for each possible value x ≈ � x , Go Back – to what extent x is possible. Full Screen Close Quit

Need for Data Processing 5. Gauging Expert Uncertainty (cont-d) Need for Expert Estimates Need to Take . . . • For example, we can ask the expert to mark her cer- Gauging Expert . . . tainty by a mark m on a scale from 0 to s . Processing Expert . . . • Then we take m/s as the degree. Interval and Fuzzy . . . Case Study: Heat Meter • The function µ ( x ) assigning degree to a value x is Acknowledgments known as a fuzzy set . Home Page • If for each variable x i , we only know that x i ∈ x i = Title Page [ x i , x i ], then we know that ◭◭ ◮◮ def y = f ( x 1 , . . . , x n ) ∈ y = f ( x 1 , . . . , x n ) = ◭ ◮ { f ( x 1 , . . . , x n ) : x i ∈ x i } . Page 6 of 13 • Computing such a range y is one of the main problems Go Back of interval computations . Full Screen Close Quit

Need for Data Processing 6. Processing Expert Uncertainty Need for Expert Estimates Need to Take . . . • For expert estimates, it is reasonable to consider: Gauging Expert . . . – for every α ∈ [0 , 1], Processing Expert . . . – the set x i ( α ) = { x i : µ i ( x i ) ≥ α } of sufficiently Interval and Fuzzy . . . possible values. Case Study: Heat Meter Acknowledgments • Then, for every α , we compute the range Home Page y ( α ) = f ( x 1 ( α ) , . . . , x n ( α )) . Title Page • This can also be done by interval computation tech- ◭◭ ◮◮ niques. ◭ ◮ • Additional problems: Page 7 of 13 – sometimes, the dependence y = f ( x 1 , . . . , x n ) is not Go Back known exactly; Full Screen – even when we know the exact dependence, we can Close often only compute f ( x 1 , . . . , x n ) approximately. Quit

Need for Data Processing 7. Processing Expert Uncertainty (cont-d) Need for Expert Estimates Need to Take . . . • The approximate character of computing f ( x 1 , . . . , x n ) Gauging Expert . . . is caused by: Processing Expert . . . – rounding errors for arithmetic operations, Interval and Fuzzy . . . – inevitably imprecise formulas for non-arithmetic el- Case Study: Heat Meter ementary functions such as exp( x ) etc. Acknowledgments Home Page • One of the main objectives of metrology is: Title Page – to provide guaranteed information about the actual ◭◭ ◮◮ values of the quantities of interest ◭ ◮ – based on measurement results and expert esti- mates. Page 8 of 13 Go Back Full Screen Close Quit

Need for Data Processing 8. Interval and Fuzzy Computations in Metrol- Need for Expert Estimates ogy: A Brief History Need to Take . . . Gauging Expert . . . • 1960s: IFIP (led by Wilkinson) proposes: Processing Expert . . . – accompanying each data processing software Interval and Fuzzy . . . – with bounds (interval) estimate of the result’s in- Case Study: Heat Meter accuracy. Acknowledgments Home Page • 1960s: Moore et al. proposed general interval tech- niques for such estimates. Title Page • 1970s: software packages with guaranteed bounds (e.g., ◭◭ ◮◮ Linpack). ◭ ◮ • 1965: fuzzy sets introduced by Zadeh. Page 9 of 13 • 1980s: L. K. Reznik combined expert estimates with Go Back measurement intervals in practical problems. Full Screen • 1985: first standard for metrological support of data Close processing. Quit

Need for Data Processing 9. Interval and Fuzzy Computations in Metrol- Need for Expert Estimates ogy: A Brief History (cont-d) Need to Take . . . Gauging Expert . . . • 1985: systematic way of providing such support de- Processing Expert . . . scribed in a special issue of Measuring Techniques . Interval and Fuzzy . . . • 1990s: further theoretical development and algorithms Case Study: Heat Meter design. Acknowledgments Home Page • 2000s–2010s: metrological proposals for taking interval and fuzzy uncertainty into account. Title Page • What we would like: to incorporate interval and fuzzy ◭◭ ◮◮ techniques in metrological practice. ◭ ◮ • What is needed for this: Page 10 of 13 – add interval and fuzzy computations to the existing Go Back metrological standards, Full Screen – make the corresponding algorithms as simple as Close possible and as clear to engineers as possible. Quit

Need for Data Processing 10. Case Study: Heat Meter Need for Expert Estimates Need to Take . . . • In many practical situations, we need to know how Gauging Expert . . . much heat or cooling was generated or consumed. Processing Expert . . . • For example, in nuclear power stations: Interval and Fuzzy . . . – water or gas is heated by a reactor, Case Study: Heat Meter Acknowledgments – the steam is moved to a turbine that generates elec- Home Page tricity, – when the steam rotates the turbine, it loses energy Title Page and cools down. ◭◭ ◮◮ • Similarly, in heating and air conditioning systems: ◭ ◮ – hot water is circulated, heating a building; Page 11 of 13 – cold air is circulated, cooling the building; Go Back – in dry areas, water is used to cool the buildings. Full Screen • In all these cases, it is desirable to measure the amount Close of heat. Quit